# Finite Element Method and Applications

## Book summary

Summary of the book…

## Book cover

Foreword

1. Functional Analysis Itinerary
1.1. Vector spaces.
1.2. Topological spaces. Topological vector spaces.
1.3. Normed vector spaces. Banach and Hilbert spaces.
1.4. Operators and imbeddings. Dual spaces.
1.5. Continuous functions spaces.
1.6. Lebesgue measure and integral in Rⁿ.
1.7. Lp  spaces and generalized derivatives.
1.8. Sobolev spaces, imbedding theorems, Friedrichs-Poincaré inequalities. Concept of trace and Green formulas.
1.9. Distributions, fundamental solutions and Green functions.
1.10. Linear variational problems. Lax-Milgram lemma.
1.11. Elements of differential calculus in normed spaces.
1.12. Elements of affine geometry.
2. The Finite Element Method (FEM) for Elliptic Equations
2.1. Linear variational problems-abstract formulations.
2.2. Variational formulation for second order problems.
2.3. Variational formulation for fourth order problems.
2.4. The bidimensional Stokes problem.
2.5. The approximation of elliptic variational problems-general theory.
2.6. The finite element method and its mathematical frame.
2.7. Multidimensional Lagrange and Hermite interpolation.
2.8. Interpolation polynomials (examples).
2.9. The notion of finite element. Fundamental properties.
2.10. The interpolation theory in Sobolev spaces.
2.11. Second order problems in polygonal domains.
2.12. Numerical integration in FEM.
2.13. Isoparametric finite elements.
2.14. Second order problems over curved domains.
2.15. The approximation of fourth order problems.
3. Finite Element-Finite Differences Method (FDM) for Parabolic Equations
3.1. The Cauchy-Dirichlet problem for heat equation-a variational formulation.
3.2. Solving the variational problem.
3.3. Error estimations and convergence for various discretization schemes.
3.4. The efficiency of Crank-Nicolson and Calahan schemes.
4. Boundary Element Method (BEM)
4.1. Introductory notions.
4.2. Boundary element method for potential problems. Indirect formulation. Direct formulation. Two-dimensional problems. Linear elements. Quadratic and higher-order elements.
4.3. Nonpotential problems treated by boundary element method. Diffusion equation. Elastostatics equations (Navier). Euler’s equations. Stokes model. Navier-Stokes equation.
4.4. Coupling of BEM and FEM.
5. General Comments on FEM Implementation and Numerical Examples
5.1. Strategies of FEM implementation
5.2. Comparison between FEM and BEM
5.3. Comparison between FEM, FDM and spectral methods (SM).
5.4. Numerical examples.

References

Contents (in Russian)

keyword1,

pdf file

## References

see the expanding block below

##### Cite this book as:

T. Petrila, C.-I. Gheorghiu, Finite Element Method and Applications, Editura Academiei Republicii Socialiste România, București, 1987 (in Romanian).

##### Book Title

Finite Element Method and Applications

##### Authors

T. Petrila, C.-I. Gheorghiu

##### Topics

[1] Atkinson, K.E., A survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind, SIAM, Philadelphia, 1976
[2] Atkinson, K.E., The numerical solution of Fredholm integral equation of second kind,  SIAM, Numer. Anal.4 (1967), pp. 337-348.
[4] Akyuz, F.A., Natural Coordinate Systems – An Automatic Input Data Generation Scheme for  FEM, Nuclear Engineering and Design, 11, (1970).
[5] Arantes E Oliveira, E.R.,  Convergence of Finite Element Solutions in Viscous Flow Problems,  IJNME*, 9 (1975), pp. 739-763.
[6] Aris, R., Vectors, Tensors and Basic Equations of the Fluid Mechanics,  Prentice Hall, New Jersey, 1962, Cap. X.
[7] Aubin, J.P., Behavior of the error of the approximate solutions of boundary value problems for linear elliptic operators by Bakerkin’s and finite difference methods, Ann. Scuola Norm, Sup. Pisa, 21 (1967), pp. 599-637
[8] Aral, M.M., Gulcat, U., A Finite Element Laplace Transform Solution Technique for  the Vave Equation,  IJNME*, 11 (1977), pp. 1719-1732
[9] Argyris, J.H., Mareczek, G., Scharpf, D.W.,  Two and Three Dimensional Flow Analysis Usign Finite Elements,  Nuclear Engineering and Design, 11(1970).
[10] Abramowitz, M., Stegun, I.A., Handbook of Mathematical functions,  Dover, New York, 1965.
[11] Babuska, I., Error bound for the finite element method,  Num. Math. 16(1971, pp.322-333.
[12] Babuska, I., The finite element method with Lagrangian multipliers, Num. Math. 20 (1973), pp. 179-192.
[13] Brezzi, F., On the existence, uniqueness and approximaiton of saddle-point problems arising from lagrangian mujltipliers. RAIRO, august 1974, R2, pp. 129-171.
[14] Baker, A.J., Finite Element Solution Alorithm for Viscous Incompressible Fluid Dynamics, IJNME*, 8(1970), pp.61-71.
[15] Baker, A.J., A Finite Element Computational Theory for the Mechanics and Thermodynamics of a Viscous Compressible Multi-Species Fluid, Bell Aerospace Research Report 9500-920200(1971).
[16] Begis, D., Perronnet, I., The Club MODULEF – A Library of Computer Procedures for Finite Element Analysis,   INRIA** – France, Aprile 1982.
[17] Bradeanu, D., Description of the finite element method with spline functions on a simple bilocal problem.  Preprint no.1, 1982, Seminar of Numerical Approximation Methods in Hydrodynamics and Heat Transfer, Univ. Cluj-Napoca.
[18]  Bradeanu, D.,  A variational method with local potential for the Mises problem,  Preprint no.1, 1982, Univ. Cluj-Napoca.
[19] Bradeanu, D., The mathematical considerations of the hydrodynamic principle of  Lord Kelvin,  Preprint, no.1, 1982, Univ. Cluj-Napoca.
[20] Bristeau, M.O., Pironneau, O., Glowinski, R., Periaux, J., Perrier, P., On the Numerical Solution of Nonlinear Problems in Fluid Dynamics by Least Square and FEM (I) Least Square Formulations and Conjugate Gardient Solution of the Continuous Problems,  CMAME***, 17/18(1979), pp. 619-657.
[21] Bratianu, C., Metode cu elemente finite în dinamica fluidelor. Edit,. Academiei, R.S.R., București 1983.
[22] Browder, F.E., On the Regularity Properties of Solutions of Elliptic Differential Equations, Comm. Pure. Math., IX(1956), pp. 351-361.
[23] Buell, W.R., Bush, B.A.,  Mesh Generation A Survey,  Trans. of ASME, Ser. B, 95(1973), 1.
[24] Bramble, J.H., Hilbert, S.R.,  Estimations of linear functional on  Sobolev spaces with application to Fourier transforms and spline interpolation, SIAM J.  Numer. Anal. 7(1970), pp.113-124.
[25] Bramble, J.H., Hilbert, S.R., Bounds for a class of linear functionals with applications to Hermite interpolation, Numer. Math., 16(1971), pp. 362-369.
[26] Bramble, J.H., Zlamal, M., Triangular elements in FEM, Math. Comp., 24 (1970), pp.809-820.
[27] Barrett, G., Demunski, G., Non-Self-Adjoint Problems and Essential Boundary Conditions, IJNME*, 14(1979), PP. 507-513.
[28] Barret, D., Demunski, G., Finite Element  Solutions of Convective Diffusion Problems, IJNME, 14(1979), PP. 1511-1524.
[29] Babuska, I., The Rate of Convergence for FEM, SIAM J. Numer. Anal.,8(1971), 2.
[30] Bramble, J.H., Nitsche, J.A.,  A Generalized Ritz-Least-Square Method for Dirichlet Problems, SIAM J. Numer. Anal., 10(1973), 1
[31] Brebbia, C.A., The Boundary Element Method for Engineers,  Pentech Press, London, Halstead press, New York, 1978.
[32] Berbbia, C.A., Walker, S., Boundary Element Techniques in  Engineering,  Butterworths, London, 1980.
[33] Brebbia, C.A., Telles, J.C.F., Wrobel, L.C.,  Boundary Element Techniques,  Springer Verlag, Berlin, Heidelberg, New York, Tokyo, 1984.
[34] Cea, A., Approximation variationnelle des problems aux limites, Ann. Inst. Fourier,  Grenoble, 14(1964), pp. 345-444.
[35] Cheng, R.T.S., Numerical Solution of Navier-Stokes Equations by FEM,  The Physics of Fluides, 15(1972), pp. 2098-2105.
[36] Cheng, R.T.S., On the Accuracy of Certain C Continuous Finite Element Representations,  IJNME, 8(1974), pp. 649-657.
[37] Cheng, Y.K., Yeo. M.E., A Practical Introduction to Finite Element Analysis , {itman, 1079.
[38] Chifu, E., Gheorghiu, C.I., Stan, I., Surface Mobility of Surfactant Solutions XI. Numerical Analysis for the Marangoni and Grawity Flow in a Thin Liquid Layer of
Triangular Section,
Rev. Roumaine Chim., 29(1984), 1
[39] Chung, T.J., Finite Element Analysis in Fluid Dynamics, Mc Graw Hill, New York, 1978.
[40] Ciarlet, P.G., Numerical Analysis of the Finite Element Method, Seminaire de Mathematiques Superieures, Department de Mathematiques – Univ. de M ontreal, 1976.
[41] Ciarlet, P.G., The Finite Element Method for Elliptic Problems, North-Holland Publishing Company, 1978.
[42] Ciarlet, P.G., Glowinski, R., Dual iterative techniques for solving a finite element approximation of the biharmonic equation, CMAME***, 5(1975), pp. 277-295.
[43] Ciarlet, P.G., Raviart, P.A.,  A mixed FEM for the biharmonic equation, in Mathematical Aspects of Finite Elements in Partial Differential Equations, C. de Boor, ed. Academic Press, New York, 1974, pp.  125-145.
[44] Ciarlet, P.G., Raviart, P.A., General Lagrange and Hermite Interpolation in Rⁿ  with Applications to FEM, Arch. Rational Mech. Anal., 46(1972), pp. 177-199.
[45] Ciarlet, P.G., Wagschal, C., Multipoint Taylor Formulas and Applications to the FEM, Numer. Math., 17 (1971), pp. 84-100.
[46] Connor, J.J., Brebbia, C.S., Finite Element Techniques for Fluid Flow,  Newnes – Buttersworths, 1977.
[47] Ciarlet, P.G., Raviart, P.A., Interpolation theory over curved elements, with applications to FEM, CMAME***, 1(1972), pp. 217-249.
[48] Ciarlet, P.G., Raviart, P.A., The combined effect of curved boundarics and numerical integration in isoparametric FEM in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations,  A.K. AZIZ, ed. Academic Press, New York, 1974, pp. 409-474.
[49] Cristescu, R., Analiză funcțională,  Edit. Didactică și Pedagogică, 1979.
[50] Cea, J., Sur un test de convergence, Journees Elements Finis, Univ. de Rennes, mai 1976.
[51] Cruse, T.A., Rizzo, F.J.,  A direct formulation and numerical solution of the generation transient elasto-dynamic problem,  I.J. Math. Anal. Appl., 22 (1968(, pp. 244-259.
[52]  Coleman, C.J., Q. Jl.  Mech. Appl. Math., 34, 4 (1981), pp. 433.
[53] Curran, D.A.S., Cross, M., Lewis, B.A., Solution of parabolic differential equations by the BEM using aiscretization in time,  Appl. Math. Modelling 4(1980(, pp. 398-400.
[54] Dincă, G., Metode variaționale și aplicații,  Editura Tehnică, București, 1980.
[55] Douglas, J. JR., Dupont, T., Galerkin Methods for Parablic Equations, SIAM, J. Numer. Anal., 7(1970), pp. 575-626.
[56] Dupont, T., Fairweather, G., Johnson, P.J., Three – Level Galerkin Methods for Parabolic Equations, SIAM J. Numer. Anal., 11 (1974), pp. 392-410.
[57] Dragoș, L., Principiile mecanicii mediilor continue,  Editura Tehnică, București, 1982.
[58] Dieudonne, J., Foundations of Modern Analysis, Academic Press, 1960.
[59]  Dunford, N., Schwartz, J.T., Linear Operators, Part. I, General Theory, Interscience Publishers, Inc.New York, 1958.
[60] Decloux, J., Methode des Elements Finis, Department de Mathematiques, Lausanne, Suisse, 1973.
[61] Douglas, J. JR., Dupont, T.,  A Finite Element Collocation Method for Quasilinear Parabolic Equations, Mathematics of Computation, 27(1973), 121
[62] Dupont, T., L²-Estimates for Galerkin Methods for Second Order Hyperbolic Equations, SIAM J. Numer. Anal., 10(1973), 5
[63] Dupont, T., Galerkin Methods for First Order Hyperbolics, An Example, SIAM J Numer. Anal. 10(1973), 10.
[64] Eckart, C.,  Variational Principles of Hydrodynamics, Phys. Fluids, 3(1960), pp. 421-427.
[65] Ekeland, I., Temam, R., Analyse Convexe et Problemes Variationnels, Dunod Gauthier-Villars, Paris, 1974.
[66] Fichera, G., Sul problema della derivata obliqua e sul problema misto per l’equazione di Laplace,  Bolll. U.M.I., III, VII(1962), pp. 367-377.
[67] Finlayson, B.A., the Method of Weithed Residuals and Variational Principles, Academic Press, 1972.
[68] Finlayson, B.A.,  Existence of Variational Principles for the Navier-Stokes Equaiton,  The Physics of Fluids, 15, 1972, pp. 963-967.
[69] Finlayson, B.A., scriven, L., The method of weithed residuals – a review,  Appl. Mech. Rev., 19 (1966), pp. 735.
[70] Finlayson, B.A., Scriven, L.E., On the search for variational principles,  International Journal Heat and Mass Transfer, 10 (1967), pp. 799-821.
[71] Fletcher, C.A.J., Computational Galerkin Methods,  Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1984.
[72] Fortin, M., Peyret, R., Temam, R., Resolution numerique des equations de Navier-Stokes pour un fluia incompressible,  Journal de Mecanique, 10 (1971), 3.
[73] Fortin, M., Thomasset, F.,  Mixed FEM for Incompressible Flow Problems,  J. Computational Physics, 31 (1979), pp. 113-145.
[74] Fichera, G., Existence theorems in elasticity – Boundary value problems of elasticity with unilateral constraints,  in Mechanics of Solids II, Ed. C. Truesdell, Springer Verlag, Berlin, 1972, pp. 347-414.
[75] Fried, I., Finite Element Analysis of Time-Dependent Phenomena,  AIAA j., 7(1979), 6.
[76] Fredholm, I., Sur une classe d’equations fonctionelles,  Acta Math. ,27 (1903), pp. 365-390.
[77] Gelfand, I.M., Șilov, G.E., Funcții generalizate,  Editura științifică și enciclopedică, București, 1983.
[78] Gallagher, R.H., Oden, J.T., Taylor, C., Zeinkiewicz, O.C., Finite Elements in Fluids, John Wiley & Sons, New York/London/Sydnay/Toronto, 1975.
[79] Gheorghiu, C.I.,  FEM for General Quasi Harmonic Equation, Preprint no.1, 1982, Univ. Cluj-Napoca.
[80] Gheorghiu, C.I., MEF în probleme  ale  mișcării fluidelor vîscoase, Teză doctorat Univ. București, decembrie 1984.
[81] Girault, V., Raviart, P.A., Finite Element Approximations of the Navier-Stokes Equations,  Lecture Notes in Mathematics 749, Springer Verlag, 1979.
[82] Girault, V, Raviart, P.A., An Analysis of Upwind Schemes for the Navier-Stokes Equations, I SIAM J. Numer. Anal., 19 (1982), 2.
[83] Glowinski, R., Approximation Externes, par Elements Finis de Lagrange d’Orde Un et Deux, du Probleme de Dirichlet pour l’Operateur Biharmonique. Metode Iterative de Resolution des Problemes Approaches, in Topics in Numerical Analysis, J.J. Miler, Academic Press, London, 1973, pp. 123-171.
[84] Glowinski, R., Pironneau, O., Numerical  Methods for First Biharmonic Equation and for two Dimensional Stokes Problem, SIAM Review, 21 (1979), pp. 167-212.
[85] Glowinski, R., Lions, J.L., Tremolieres, R.,  analyse Numerique des Inequations Variationnelles,  Tome 1 et 2, Bordas, Paris, 1976.
[86] Galbură, Gh., Rado, F.,  Geometrie,  Edit. Didactică și Pedagogică, București, 1979.
[87] Greene, B.E., Jones, R.E., McLay, R.W., sgtrome, D.R., Generalized Variational Principles in the FEM,  AIAA J., 7(1969), 7.
[88] Gheorghiu, C.I., Stan, I., Unele aspecte ale curgerii lichidelor pe un plan înclinat în prezența gradienților de tensiune superficială. Colocviul național de Mecanica Fluidelor și aplicațiile ei tehnice, Constanța, 1980.
[89] Gheorghiu, C.I., Calculul mișcărilor fluide vîscoase peste praguri folosind MEF.  Colocviul național de Mecanica Fluidelor și aplicațiile ei tehnice, Galați, 1982.
[90] Gagliardo, E., Caracterizzatione delle tracce sulla frontiera relative ad alcune classi di funzioni in n  variabili. Rend. Sem. Mat. Padova, 27 (1957), pp. 284-305.
[91] Hildebrandt, St., Wienholts, E.,  Constructive proofs of representations theorem in separable Hilbert space,  Comm. Pure Appl. Math., 17 (1964), pp. 369-373.
[92] Haber, S.,  Numerical evaluation of multiple integrals,  SIAM Rev., 12 (1970), pp. 481-526.
[93] Huebner, K.H.,  The FEM for Engineers,  John Wiley & Sons, 1975.
[94] Hartmann, F.,  Computing the C-matrix in nonsmooth boundary points, in New Developments in Boundary Element Methods (C.A. Brebbia, Ed.), pp. 367-379, Butterworths, London, 1980.
[95] Helmholtz, H., Theorie der Luft schwingungen in Rohren mit offenen Enden,  Crelle’s J., 57 (1960), pp. 1-72.
[96] Hsiao, G.C., Wendland, W.L., Super approximation for boundary integral methods,  Proc. of the Fourth IMACS Conf. , 1981.
[97] Iacob, C., Introduction Mathematique a la Mechanique des Fluides,  Edit. Academiei R.P.R. si Gauthier-Villars, Paris, 1959.
[98] Iacob, C., Determination de la seconde approximation de l’ecoulement compressible subsonique autour d’un profil donne,  Archivum Mechaniki Stosowanej 16 (1964), 2.
[99] Irons, B.M.,  A Frontal Solution Program for Finite Element Analysis, IJNME*, 12 (1970), pp. 5-32.
[100] Irons, B.M., Razzaque, A.,  Experience with patch – test for convergence of finite elements in The Mathematical Foundation of the FEM with Applications to PDE, A. K. AZIZ ed., Academic Press, New York, 1972.
[101] Ingham, D.B., Kelmanson, M.A., Boundary Integral Equation. Analyses of Singular Potential and Biharmonic Problems,  Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1984.
[102] Jaswon, M.A. and Symm, G.T., Integral Equation Methods in Potential Theory and Elastostatics,  Academic Press, London, 1977.
[103] Kalik, C., Ecuații cu derivate parțiale,  Editura Didactică și Pedagogică, București, 1980.
[104] Kalik, C., Ecuații cu derivate parțiale II – Metoda elementului finit,  Lito. Univ. Cluj-Napoca.
[105] Kawahara, M., Yoshimura, N., Nakagawam K., Ohsaka, H.,  Steady and Unsteady Finite Element, Analysis of Incompressib le Viscous Fluid, IJNME*, 10 (1976)
[106] Kolmogorov, A., Fomine, S., Elements de la Theorie des Fonctions et de lțAnalyse Fonctionnelles,  Ed. Mir., Moscou, 1974.
[107] Kirillov, A., Gvichiani, A., Theoremes et problemes dțanalyse fonctionnelle, Ed. Mir., Moscou, 1982.
[108] Krahula, J.L., Polhemus, J.F., Use of Fourier Series in the FEM, AIAA J., 6(1968), 4
[109] Kikuchi, F., A FEM for Non-Self-Adjoint Problems, IJNME*, 6 (1973), pp.39-54.
[110] Kellogg, O.D., Foundation of Potential Theory, Springer Verlag, Berlin, 1929.
[111] Kirchoff, G., Zur Theorie der Lichstrahlen,  Berl. Ber., (1882), pp. 641.
[112] Kupradze, O.D., Potential methods in the theory of elasticity, Daniel Davey & Co., New York, 1965.
[113] Kutt, H.R., The numerical evaluation of principal  value integrals by finite part integration, Numer. Math., 24 (1975), pp. 205-210.
[114] Kufner, A, John, O., Fucik, S., Function spaces, Ed. Acad. Publ. H, Prague, 1977.
[115] Ladyzenskaja, O.A.,  Linejnie i kvazilinejnye uravnenija elipticekogo typa.  Nauka, Moskva, 1964.
[116] Ladyzsenskaja, O.A., Linejnie i kvazilinejnye uravnenija paraboliceskogo typa. Nauka, Moskva, 1967.
[117] Lions, J.L., Pr oblemes dux limites dans les equations aux derivees partielles, Les Presses de l’Universite de Montreal, Sept. 1965.
[118] Lions, J.L., Quelques Methodes de Resolution des Problemes aux Limites Non Lineatres, Dunod, Paris, 1969.
[119] Lions, J.L., Magenes, E., Non-Homogeneous Boundary Value Problemes and Applications, Springer-Verlag, 1972.
[120] Lynn, P.P., Least Squares Finite Element Analysis of Laminar Bolundary Layer Flow, IJNME, 8 (1974), pp. 865-876.
[121] Lynn, P.P., Arya, S.K., Use of the Least Squares, Criterion in the Finite Element Formulation,  IJNME*, 6 (1973), pp. 75-88.
[122] Lions, J.L., stampacchia, G., Variational Inequalities, Comm. Pure Appl. Math. 20 (1967), pp. 493-519.
[123] Landau, L., Lifchitz, E., Theorie de l’Elasticite, Mir. Moscou, 1967.|
[124] Love, A.E.H., A Treatise on the Mathematical Theory of Elasticity, Dover, New York, 1944.
[125] Ladyzhenskaya, O.A.,  The mathematical theory of viscous incompressible flow, Gordon and Breach, New York, 1963.
[126] Lighthill, M.L., Introduction Boundary layer theory, in Laminar Boundary Layer (L. Rosenhend, Ed.), Oxford University Press London, 1963.
[127] Magenes, E., Recente Sviluppi nella teoria dei problem misti per le  equazioni lineari  ellittiche, Rendiconti del Seminario Matematico e Fisico di Milano XXVII, (1956).
[128] Marciuk, G.I., Metode de analiză numerică, traducere din limba rusă, Edit. Academiei R.S.R., 1983.
[129] Marcov, N., Metoda elementelor finite, in Matematici Clasice și Moderne,  vol. IV,  (coord. acad. C. Iacob_ Edit. Tehnică, București, 1984.
[130] Merreifeild, B.C., Fortran,  Subroutines for Finite Element Analysis, RAE Technical Report 71156 – Arc 33734.
[131] Mihlin, S.G., Smolitskiy, K.L., Approximate Methods for Solutions of Differential and Integral Equations, American Elsevier Publishing Company Inc., New York, 1967.
[132] Mihlin, S.G., Ecuații liniare cu derivate parțiale, Editura Științifică și enciclopedică, București, 1983.
[133] Miranda, C., Sul problema misto per le equazioni lineari ellittiche, Annali di Matematica Pura ed Aplicata, Ser. IV, XXXIX (1955)
[134] Marinescu, G., Tratat de analiză funcțională,  Editura Academiei R.S.R., București, vol. I, 1970, vol. II. 1972.
[135] Marinescu, G., Analiză numerică, Editura Academiei R.S.R., București, 1974.
[136] Micula, G., Funcții spline, Editura Tehnică, București, 1978
[137] Mayer, H.D., The Numerical Solution of Nonlinear Parabolic Problems by Variational Methods,  SIAM J. Numer. Anal. 10 (1973).
[138] Mindlin, R.D., Force at a point in the interior of a emiinfinite solid, Physica, 7 (1936), pp. 195-202
[139] Mihăilescu, M., Gheorghiu, C.I., Mariș, I.,  Theoretical and experimental comparative analysis of perforated beams,  International Conference on Computer – Aided Analysis and Design of Concrete  Structures, Split, Yugoslavia, 1984.
[140] Martin, H.C.,  Finite element analysis of fluid flows, Proceedings 2 nd Conference on Matrix Methods in Structural Mechanics, AFFDL-TR-68-150 (1969)
[141] Muskhelishvili, N.I.,  Some basic problems of the mathematical theory of elasticity,  Nordhoff, Gromingen, 1953.
[142] Mihlin, S.G.,  Integral equations, Pergamon, New York, 1957.
[143] Morse, P.M., Feshbach, H., Methods of Theoretical Pgysics,  Mc. Graw-Hill, New York, 1953.
[144] Mihlin, S.G., Variationsmethoden der Mathematischen Physik, Akademic-Verlag, Berlin, 1962.
[145] Norrie, D.H., de Vries, G., The Finite Element Method (Fundamental and Applications), Academic Press, New York, 1973.
[146] Norrie, D.H., de Vries, G., An Introduction to Finite Element Analysis, Academic Press, New York, 1978.
[147] Necas, J., Les Methodes Directes en Theorie des Equations Elliptiques,  Masson, Paris, 1967.
[148] Nicolaides, R.A., On the Class of Finite Elements Generated by Lagrange Interpolation, SIAM J. Numer. Anal., 9 (1972), pp. 435-445.
[149] Nitsche, J., Ein kriterium fur die quasi-optimalitat des Ritschen Verfahrens,  Numer. Math., 11 (1968), pp. 346-348.
[150] Neumann, C., Untersunchungen uber das logarithmische und Newtonsche potential,  Teubner, Leipzig, 1877.
[151] Oden, J.T.,  Finite Elements of Nonlinear Continuu, Mc Graw Hill Book Company, 1972.
[152] Oden, J.T., Reddy, J.N., On dual complementary variational principles in mathematical physics, Int. J. Engn. Sci. 12 (1974), pp. 1-29.
[153] Oden, J.T., Reddy, J.N., An Introduction to the Mathematical Theory of Finite Elements, John Wiley & Sons, 1976.
[154] Oroveanu, T., Mecanica fluidelor vîscoase, Editura Academiei R.S.R., București, 1967.
[155] Olson, M.D., Variational – FEM for Two-Dimensional and Axisymmetric Navier Stokes Equations,  in Finite Elements in Fluides,  Vol. I, Ed. by Gallagher, Oden, Tayls, Zienkiewicz, John & Sons, 1975.
[156] Periaux, J., Three Dimensional Analysis of Compressible Potential Flows with the FEM,  IJNME, 19 (1975), pp. 775-831.
[157] Petrila, T.,  Modele matematice în hidrodinamica plană, Editura Academiei R.S.R., București, 1981.
[158] Peyret, R., Taylor, T.D.,  Computational Methods for Fluid Flow, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1984.
[159] Pin Tong, Simplex Elements of C⁰  Continuity with Varging Polinomial Degrees,  IJNME*, 11, (1977), pp. 27-38.
[160] Popescu, I.P., Geometrie afină  și euclidiană, Editura Facla, Timișoara, 1984.
[161] Prenter, P.M., Splines and Variational Methods, John Wiley & Sons, 1975.
[162] Petrila, T.,  Mouvement general d’un profil dans un fluid ideal en presence d’une paroi permeable illimitee. Cadre variationnel  et approximation par une methode d’elements finis,  Mathematica-Revue d’Analyse numerique et la theorie de l’approximation, 8, 1, (1979), pp. 67-77.
[163] Petrila, T., Gheorghiu, C.I.,  Metoda elementelor finite pe frontieră în unele probleme ale matematicii fluidelor vîscoase,  Colocviul național de Mecanica Fluidelor și aplicațiile ei tehnice, Mediaș, 1985.
[164] Petrila, T., Gheorghiu, C.I., Considerations on the internal MEF for Stokes model, Proceedings of the Conference on Differential equations, Cluj-Napoca, 21-23 noiembrie, 1985.
[165] Pavel, P., Rus, A.I.,  Ecuații diferențiale și integrale, Editura didactică și pedagogică, București, 1975.
[166] Petrila, T., An improved CVBEM for plane hydrodynamics, Proceedings of the Seminar on Complex Analysis, Cluj-Napoca, 20-22, noiembrie, 1986.
[167] Roache, P.J., Computational Fluid Dynamics, Hermosa Publishers, Albuquerque, N.M., 1972.
[168] Risso, F.J., Shippu, D.J., An advanced boundary integral equation method for three dimensional thermoelasticity, IJNME*, 11, pp. 1753-1768 (1977).
[169] Rayleigh, Lord,  Theory of Sound, Dover, NYC, New York, 1887.
[170] Risso, F.J., Shippy, D.J., A method of solution for certain problems of transient, heat conduction, AIAA J. 8 (1970), pp. 2004-2009.
[171] Richardson, S.,  Proc. Camb. Phil. Soc., 67 (1970), pp. 47
[171] Schechter, E., Metoda elementului finit,  Curs la Facultatea de Matematică, Univ. Cluj-Napoca, 1981-1982.
[172] Schechter, M., Mixed Boundary Problems for General Elliptic Equations, Comm. Pure. Appl. Math. XIII(1960), pp. 183-201.
[173] Schechtger, M., Mixed Boundary Problems for General Elliptic Equations,  Comm.  Pure. Appl. Math. XIII (1960), pp. 183-201.
[174] Segerlind, L.J., Applied Finite Element Analysis, John & Sons, 1976.
[175] Seliger, R.L., Whitman, G.B., Variational Principles in Cotinuum Mechanics, Proc. Roy. Soc., A305 (1968), pp. 1-25.
[176] Stampacchia, G., Problemi al contorno misti per equazioni del calcolo delle variazioni, Anali di Matematica Pura ed Applicata, Ser. IV, XI, Bologna, 1955.
[177] Syres, J., Rae, J., An Introduction to the Use of the FEM in Flow Modelling,  Theoretical Physics Division AERE, Harwell, Oxfordshire, June, 1976.
[178] Strang, G., Variational crimes in the FEM, in The Mathematical Foundations of the FEM with Applications to PDE,  A.K. AZIZ, ed. Academic Press, New York, 1972, pp.689-710
[179] Strang, F., Fix, G.J., An Analysis of the FEM,  Prentice Hall, Englewood Cliffs, 1973.
[180] Somigliana, C.,  Sopra l’equilibrio di un corpo elastico isotropo,  Il Nuovo Ciemento 17-19 )1886)
[181] Schapery, R.A., Approximate methods of transform inversion for visco-elastic stress analysis, Proc. Fourth U.S. National Congress on Applied Mechanics, Vol. 2, 1962.
[182] Szerget, P., Alujevic, A., The solutions of Navier-Stokes equations in terms of vorticity-velocity variable by the BEM, ZAMM, 65 (4) (1985), pp., 245-248.
[183] Temam, R.,  Metode numerice de rezolvare a ecuațiilor funcționale, Editura Tehnică, București, 1973.
[184] Thomasset, F., Equations de Navier-Stokes Bidimensionnelles – Modules NSNCEV – NSNCST, TRSD; INRIA** – Laboria, Nov. 1979.
[185] Thomasset, F., Implementation of  FEM for Navier-Stokes Equations,
Springer-Verlag, 1981.
[186] Tottenham, H., A Direct Numerical Method for the Solution of Field Problems, IJNME, 2 (1970), pp. 117-131.
[187] Tay, O.A., Vahl Davis, G., de.,  Application of the FEM to Convection Heal Transfer Between Parallel Planes, Int. Heat Mass Transfer, 14 (1971), pp. 1057-1069.
[188] Treves, F.,  Introduction to Pseudodifferential and Fourier Integral Operators, I. Plenum Press, New Yorks, London, 1980.
[189] Vainberg, M.M.,  Variational Methods for the Study of Nonlinear Operators, Holden Day, San Francisco, California, 1964.
[190] Vries, G., de, Norrie, D.H., The Application of the Finite Element Techniques to Potential Flow Problems,  Transactions of the ASME, Dec. (1971).
[191] Villadsen, J.V., Stewart, W.E., Solution of boundary value problems by ortogonal collocation. Chemical Engineering Sciences, 22 (1967), pp. 1483-1501.
[192] Villadsen, J.V., Sorensen, J.P.,  Solution of parabolic PDE by a double collocation method, Chemical Engineering Science, 24 (1969), pp. 1337-1349.
[193] Veubeke, B.M., Fraijs, de., Hogge, M.A.,  Dual Analysis for Heat Conduction Problems by Finit Elements, IJNMET, 5 (1971), pp. 65-82.
[194] Vainikko, G., On the question of convergence of GALERKIN’S method,  Tartu Rukl. Ul. Tom. 177 (1965), pp. 148-152.
[195] Vladimirov, V.,  Distributions en phusique mathematique, Ed. Mir., Moscou, 1979.
[196] Zienkiewicz, O.C., Cheung, Y.K.,  The FEM in Structural and Continuum Mechanics, Mc. Graw Hill, New York, 1967.
[197] Zienkiewicz, O.C., Parech, C.J., Transient Fielt Problems: Two-Dimensional and Three-Dimensional Analysis by Isoparametric Finite-Elements, IJNME*, 2 (1970), pp. 61-71.
[198] Zienkiewicz, O.C., Phillips, D.V., An Automatic Mesh Eneration Scheme for Plane and Curved Surfaces by Isoparametric Coordinates, IJNME, 3 (1971), pp. 519-528.
[199] Zlamal, M., FEM for Parabolic Equation, Mathematics of Computation, 28 (1974), pp. 393-404.
[200] Zlamal, M., Curved Elements in the FEM I, SIAM J. Numer. Anal, 10 (1973), pp. 229-240; Part. II, 11 (1974), pp. 347-362.
[201] Zlamal, M.,  The FEM in domains with curved boundaries, IJMME*, 5 (1973), pp. 367-373.
[202] Zlamal, M., On the FEM, Numer. Math. 12 (1968), pp. 394-409.
[203] Zlamal, M.,  A finite element procedure of the second order of accuracy,  Numer. Math., 16 (1970), pp. 394-402.
[204] Zienkiewicz, O.C., Kelly, D.W., Bettes, P., The coupling of the finite element method and boundary solution procedures, Int. J. Numerical Meth. Engng., 11 (1977), pp. 355-375.
[205] Wallace, H.A.,  An Introduction to Algebraic Topology, Pergamon Press, 1967.
[206] Windder, V.D.,  The Laplace transform, Princeton University Press, Princeton, 1956.
[207] WU, J.C., Thompson, J.F., Numerical solutions of time-dependent incompressible Navier-Stokes equations using an integrodifferential formulation, Comput. Fluids 1(1973), pp. 197-215.
[208] Wendland, W.L., I Asumptotic convergence of BEM, II. Integral equation methods for mixed boundary value problems, T.H. Darmstadt, Preprint 611, 1981.
[209] Wendland, W.L., Boundary element methods and their asymptotic convergence,  T.H. Darmstadt, Preprint 690, 1982.
[210] Wenland, W.L., On some mathematical aspect of BEM for elliptic problems, T.H. Darmstadt, Preprint 857, 1984.
[211] Yosida, K., Functional Analysis, Grundlehren B., 123, Springer-Verlag, 1965.
[212] Youngren, G.K., Acrivos, A., Stokes flow post a partide of arbitrary shape. A numerical method of solution, J. Fluid. Mech. 69, part.2 )1975_, pp. 377-403.

* IJNME – International Journal for N umerical Methods in Engineering
**INRIA – Institut National de Recherche en Informatique et en Automatique (France)
***CMAME  – Computer Methods in Applied Mechanics and Engineering